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Theorem fusgrvtxfi 29204
Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)
Hypothesis
Ref Expression
isfusgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
fusgrvtxfi (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi
StepHypRef Expression
1 isfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
21isfusgr 29203 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
32simprbi 495 1 (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6549  Fincfn 8964  Vtxcvtx 28881  USGraphcusgr 29034  FinUSGraphcfusgr 29201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-fusgr 29202
This theorem is referenced by:  fusgrfupgrfs  29216  nbfusgrlevtxm1  29262  nbfusgrlevtxm2  29263  nbusgrvtxm1  29264  uvtxnm1nbgr  29289  cusgrm1rusgr  29468  wlksnfi  29790  fusgrhashclwwlkn  29961  clwwlkndivn  29962  fusgreghash2wsp  30220  numclwwlk3lem2  30266  numclwwlk4  30268
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